Answered

For 2021, the amount of U.S. federal tax owed [tex]\( T \)[/tex] is a function of taxable income [tex]\( d \)[/tex]:

[tex]\[
T(d) =
\begin{cases}
0.1d & \text{if } d \leq 9875 \\
0.12(d - 9875) + 987.5 & \text{if } 9876 \leq d \leq 40125 \\
0.22(d - 40125) + 4617.5 & \text{if } 40126 \leq d \leq 85525 \\
0.24(d - 85525) + 14605.5 & \text{if } 85526 \leq d \leq 163300 \\
0.32(d - 163300) + 33271.5 & \text{if } 163301 \leq d \leq 207350 \\
0.35(d - 207350) + 47367.5 & \text{if } 207351 \leq d \leq 518400 \\
0.37(d - 518400) + 156235 & \text{if } d \ \textgreater \ 518400
\end{cases}
\][/tex]

Noam has \[tex]$90,000 in taxable income, and Betty has \$[/tex]120,000. How much more tax does Betty pay?



Answer :

GinnyAnswer
To determine how much tax Noam and Betty owe based on their taxable incomes and the given tax brackets, let's carefully go through the calculation.

### For Noam:
Noam's taxable income is \[tex]$90,000. This falls in the fourth bracket, where \(85,526 \leq 90,000 \leq 163,300\), so we use the third condition: \[ T(d) = 0.24(d - 85,525) + 14605.5 \] Plugging in \(d = 90,000\): \[ T(90,000) = 0.24(90,000 - 85,525) + 14,605.5 \] \[ T(90,000) = 0.24(4,475) + 14,605.5 \] \[ T(90,000) = 1,074 + 14,605.5 \] \[ T(90,000) = 15,679.5 \] So, Noam owes \$[/tex]15,679.5 in taxes.

### For Betty:
Betty's taxable income is \[tex]$120,000. This falls in the fourth bracket as well, where \(85,526 \leq 120,000 \leq 163,300\), so we use the same fourth condition: \[ T(d) = 0.24(d - 85,525) + 14605.5 \] Plugging in \(d = 120,000\): \[ T(120,000) = 0.24(120,000 - 85,525) + 14,605.5 \] \[ T(120,000) = 0.24(34,475) + 14,605.5 \] \[ T(120,000) = 8,274 + 14,605.5 \] \[ T(120,000) = 22,879.5 \] So, Betty owes \$[/tex]22,879.5 in taxes.

### Difference in Tax:
To find out how much more tax Betty pays compared to Noam, we find the difference between their tax amounts:

[tex]\[ \text{Tax Difference} = T(120,000) - T(90,000) \][/tex]

[tex]\[ \text{Tax Difference} = 22,879.5 - 15,679.5 \][/tex]

[tex]\[ \text{Tax Difference} = 7,200 \][/tex]

Thus, Betty pays \$7,200 more in taxes than Noam.